Question: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}7x+2y &= 5 \\ 8x+4y &= 7\end{align*}$
Answer: Begin by moving the $y$ -term in the second equation to the right side of the equation. $8x = -4y+7$ Divide both sides by $8$ to isolate $x$ $x = {-\dfrac{1}{2}y + \dfrac{7}{8}}$ Substitute this expression for $x$ in the first equation. $7({-\dfrac{1}{2}y + \dfrac{7}{8}}) + 2y = 5$ $-\dfrac{7}{2}y + \dfrac{49}{8} + 2y = 5$ Simplify by combining terms, then solve for $y$ $-\dfrac{3}{2}y + \dfrac{49}{8} = 5$ $-\dfrac{3}{2}y = -\dfrac{9}{8}$ $y = \dfrac{3}{4}$ Substitute $\dfrac{3}{4}$ for $y$ in the top equation. $7x+2( \dfrac{3}{4}) = 5$ $7x+\dfrac{3}{2} = 5$ $7x = \dfrac{7}{2}$ $x = \dfrac{1}{2}$ The solution is $\enspace x = \dfrac{1}{2}, \enspace y = \dfrac{3}{4}$.